# Calculate: \text{begin}array l 60

## Expression: $\begin{array} { l }60,& 30,& 15,& \frac{ 15 }{ 2 },& …\end{array}$

Calculate the ratio between each pair of consecutive terms

$\begin{array} { l }r=\frac{ 30 }{ 60 },\\r=\frac{ 15 }{ 30 },\\r=\frac{ \frac{ 15 }{ 2 } }{ 15 }\end{array}$

Cancel out the common factor $30$

$\begin{array} { l }r=\frac{ 1 }{ 2 },\\r=\frac{ 15 }{ 30 },\\r=\frac{ \frac{ 15 }{ 2 } }{ 15 }\end{array}$

Cancel out the common factor $15$

$\begin{array} { l }r=\frac{ 1 }{ 2 },\\r=\frac{ 1 }{ 2 },\\r=\frac{ \frac{ 15 }{ 2 } }{ 15 }\end{array}$

Simplify the complex fraction

$\begin{array} { l }r=\frac{ 1 }{ 2 },\\r=\frac{ 1 }{ 2 },\\r=\frac{ 1 }{ 2 }\end{array}$

Since the ratio between each pair of consecutive terms is the same, the sequence is geometric and the common ratio is $r=\frac{ 1 }{ 2 }$

$r=\frac{ 1 }{ 2 }$

Substitute $\frac{ 1 }{ 2 }$ for $r$ in the recursive equation for a geometric sequence, $a_n=ra_n-1$

$a_n=\frac{ 1 }{ 2 }a_n-1$

The recursive formula consists of the first term and the recursive equation

$\begin{array} { l }a_1=60,& a_n=\frac{ 1 }{ 2 }a_n-1\end{array}$

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